A geometric construction of tight Gabor frames with multivariate compactly supported smooth windows

TL;DR

This paper presents a geometric method to construct multivariate Gabor frames with smooth, compactly supported windows by analyzing lattice fundamental domains and extending results through symplectic equivalence.

Contribution

It introduces a new geometric construction approach for Gabor frames with smooth, compactly supported windows for a broader class of lattices.

Findings

Constructed Gabor frames with smooth, compactly supported windows

Extended construction to larger lattice classes via symplectic equivalence

Utilized geometric properties of lattice fundamental domains

Abstract

The geometry of fundamental domains of lattices was used by Han and Wang to construct multivariate Gabor frames for separable lattices. We build upon their results to obtain Gabor frames with smooth and compactly supported window functions. For this purpose we study pairs of lattices which have equal density and allow for a common compact and star-shaped fundamental domain. The results are then extended to a larger class of lattices via symplectic equivalence.